Strategies of Differential Evolution for Optimum Cropping Pattern

MATERIALS AND METHODS

Farmers in the study area are allocated farmlands by the management of Orange River catchments. The study was conducted from July, 2008 to August, 2009. The allocated farmlands are 60, 70, 75 and 135 ha. The irrigation water supplied to the farms is 7 143 m³ ha^-1. The objective of the farmers is to maximize their profit using the available irrigation water maximally. The objective function of the model is formulated to maximize the TNB in monetary value derived from planting different crops on farm units.

The objective function can be expressed as:

(1)

(2)

Where:

The constraints of the problem are as follows:

The total area is equal to the sum of areas of farm units on a farmland

(3)

where, A can be 60, 70, 75 and 135 ha for each of the farms modeled.

The irrigation release for the whole year is equal to the irrigation release per hectare per year multiplied by the area of the farm in hectares. We have 4 different areas for each of the farm units.

(4)

The volume of water available is equal to the total of crop water requirement for each crop multiplied by the area where the crop is grown.

(5)

Where:

To make sure that all the crops are grown, then each area, A_i of farm units must be equal or greater than 2.5 ha. With this constraint, it is certain that all the crops will be grown in at least 2.5 ha of land. Minimum planting areas are essential to make farming survive financially. Large farming units are capable of rendering adequate financial surpluses. Models with large irrigated lands yield acceptable financial returns. In the selection of planting areas, account was taken of the maximum area which if planted nationally, would not lead to a market collapse through over-production.

(6)

Double cropping is used in this study and will help the farmers to generate more benefit from the irrigated land. Also, many crops are planted on more than an area of land.

To take account of labour requirements on the farm, the following parameters were used. For annual and fodder crops, the labour requirement is 1.3 labour days per 1000 m³. For high percentage of orchard crops, the labour requirement is 20 labour days per 1000 m³. These values are taken from a previous study (DWAF, 2004).

RESULTS AND DISCUSSION

The objective function and the constraints are solved using the 10 strategies of DE to compare the results and find out which strategy is the best for this model using the number of iterations before convergence and maximum total income generated as criteria. The total farm income is maximized, constrained by land and water availabilities. From the sensitivity analysis that was performed to determine the best combination of population size (NP), crossover constant (CR) and weighting factor (F) that is the best for this model, it was found that NP, CR and F of 40, 0.95 and 0.5, respectively are the best for the model. This is in agreement with Price and Storn (2008) that suggested that NP should be chosen as 10 times the number of parameters, CR to be slightly above or below 1.0 for fast convergence and F to be taken from 0.1 to 0.9. The model is run for each farmland of 60, 70, 75 and 135 ha. The cropping patterns are varied from A (wheat/groundnut, drybean/maize, cotton/drybean and wheat/carrot), B (wheat/potatoes, onions/watermelon, lucern and pecan) to C (wheat/groundnut, drybean/maize, olive and citrus) with the corresponding crop(s) in each pattern planted on farm units of A1, A2, A3 and A4, respectively as single or double cropping.

The 10 strategies of DE have different formulations. These formulations are presented in Table 1. Strategies 1 to 5 have binomial crossover and strategies 6 to 10 have exponential crossover. The 10 strategies are tested with the model in this study.

In Table 2, the number of iteration, total income, total area and total man-days for planting area 60 ha for each of the strategies are presented. Table 3 shows the results of cropping pattern A for farmland of 70 ha. Also the number of iteration, total income, total area, total volume and total man-days for the farmland are presented. Table 4 presents the results of different strategies of DE, iterations before convergence, total income generated, total planting areas, total volume of water, total man-days and 4 different planting areas of cropping pattern A for farmland of 75 ha. Table 5 presents the results of cropping pattern A for farmland of 135 ha. The number of iteration, total income, total area and total man-days for planting area 135 ha are presented.

Table 2:	Results of cropping pattern A for planting area 60 ha

Table 3:	Results of cropping pattern A for planting area 70 ha

Table 4:	Results of cropping pattern A for planting area 75 ha

Table 5:	Results of cropping pattern A for planting area 135 ha

Table 6:	Results of cropping pattern B for planting area 60 ha

Table 6 presents the results generated by all the strategies of DE for cropping pattern B for farmland of 60 ha. In this table, the different planting areas for the 4 different crops are presented. The total income, total area, total volume and total man-days are also presented. In Table 7, the results generated by the 10 strategies of DE for cropping pattern C for farmland of 70 ha are presented with the planting areas for each of the crops. Table 8 shows the results of cropping pattern B for farmland of 75 ha generated by the DE strategies. In this table, the total income, total area, total volume and total man-days are presented. Table 9 presents the results generated by the 10 DE strategies for cropping pattern B for farmland of 135 ha with the planting areas for each of the crops or crop combinations.

In Table 10, the results of cropping pattern C for farmland of 60 ha by the 10 strategies of DE are presented. Table 11 presents the results of cropping pattern C for farmland of 70 ha. Results of cropping pattern C for farmland of 75 ha are presented in Table 12. In Table 13, the results for the 10 different strategies of DE for cropping pattern C for farmland of 135 ha are presented. In Table 10-13, the number of iterations, total income, total area, total volume and total man-days are presented.

Table 7:	Results of cropping pattern B for planting area 70 ha

Table 8:	Results of cropping pattern B for planting area 75 ha

Table 9:	Results of cropping pattern B for planting area 135 ha

Table 10:	Results of cropping pattern C for planting area 60 ha

Figure 1 shows the study area. Figure 2 presents the maximum total incomes in South African Rand (ZAR) generated for different cropping patterns on each of the farmlands. Figure 3 shows the minimum total irrigation water for different farmlands of 60, 70, 75 and 135 ha. In Fig. 4, labour requirement in man-days for maximum profit in each of the farmlands are presented.

Table 11:	Results of cropping pattern C for planting area 70 ha

Table 12:	Results of cropping pattern C for planting area 75 ha

Table 13:	Results of cropping pattern C for planting area 135 ha

Figure 5 shows the total incomes in South African Rand (ZAR) generated from different strategies of DE for farmland 60 ha using cropping pattern C. Figure 6 shows the total incomes in South African Rand (ZAR) generated from different strategies of DE for farmland 70 ha using cropping pattern C. In Fig. 7, the total incomes in South African Rand (ZAR) generated from different strategies of DE for farmland 75 ha using cropping pattern C are presented. The total incomes in South African Rand (ZAR) generated from different strategies of DE for farmland 135 ha using cropping pattern C are presented in Fig. 8.

From Table 2, it can be found that strategy 2 (DE/best/1/bin) is the best with the lowest iterations before convergence of 153, highest total income of ZAR 889 160, maximum total area of 60 ha, minimum total volume of 512 400 m³ and lowest total man-days of 99 427. The corresponding planting areas are 17, 20, 20 and 3 ha for A1, A2, A3 and A4, respectively. Though other strategies performed well with equal total income, total area, total volume and total man-days, strategy 2 has the lowest iterations before convergence. In Table 3, strategy 2 also performed better than other strategies with only 209 iterations before convergence.

Fig. 1:	The study area

Table 4 shows that strategy 3 outperformed other strategies including strategy 2 with 190 iterations before convergence and maximum total income of ZAR 1 106 400 against ZAR 1 106 200 for strategy 2. In Table 5, strategy 2 with 208 iteration before convergence outperformed others with ZAR 1 967 100 total income. Some strategies have very high iterations before convergence like strategy 7 with 1000 iterations.

Fig. 2:	The maximum total incomes in South African Rand (ZAR) generated for different cropping patterns on each of the farmlands

Fig. 3:	The minimum total irrigation water for different farmland of 60, 70, 75 and 135 ha

Results in Table 6 shows that strategy 3 outperformed others with 161 iterations before convergence. Though strategy 9 has only 73 iterations before convergence, it fails to generate the highest total income. In Table 7, strategy 3 outperformed other strategies with maximum total income of ZAR 3 041 500 and 64 iterations before convergence. Though strategy 2 converged only after 64 iterations, it fails to generate the highest maximum total income. In Table 8, the results show that strategy 2 outperforms other strategies with 123 iterations before convergence and maximum total income of ZAR 3 246 200. Strategy 3 generated the same maximum total income but with higher iterations of 141. Results in Table 9 show that strategy 2 is the best strategy with only 81 iterations before convergence and total income of ZAR 6 067 200. Though strategies 1, 3, 4 and 5 generated the same total income, they have higher iterations of 202, 176, 236 and 388, respectively.

Fig. 4:	The labour requirements in man-days for maximum profit in each of the farmlands

Fig. 5:	The total incomes in South African Rand (ZAR) generated from different strategies of DE for farmland 60 ha using cropping pattern C

In Table 10, it is found that strategy 2 is the best DE strategy with 130 iterations before convergence and total income of ZAR 2 475 300. Strategies 1, 4 and 5 generated the same total income but higher iterations of 362, 321 and 419, respectively.

Strategy 2 with total income and iterations of ZAR 2 729 500 and 233, respectively is the best strategy in Table 11. Strategies 1, 4 and 5 have equal total income but higher iterations of 517, 529 and 771, respectively. Results in Table 12 show that DE strategy 2 with 84 iterations and ZAR 3 209 100 total income outperformed other DE strategies. Other DE strategies with binomial crossover method have the same total income but higher iterations of 191, 179, 158 and 311 respectively. This shows that DE strategies with binomial crossover method (strategies 1 to 5) are better than the strategies with exponential crossover (strategies 6 to 10) in this model. This can also be seen in the results presented. This is supported by the study by Vasan and Raju (2007) that suggest that DE/rand-to-best/1bin strategy which is also a binomial crossover method is the best giving maximum benefit taking minimum CPU time for their model.

Fig. 6:	The total incomes in South African Rand (ZAR) generated from different strategies of DE for farmland 70 ha using cropping pattern C

Fig. 7:	The total incomes in South African Rand (ZAR) generated from different strategies of DE for farmland 75 ha using cropping pattern C

In Table 13, strategy 2 with iterations of 129 and maximum total income of ZAR 5 325 900 outperforms other strategies. It can also be seen that other strategies with binomial crossover method generated the same total income with higher iterations. It can be concluded from the results in Table 2-13 that DE strategies with binomial crossover and particularly, strategy 2 is the best for this model followed by strategy 3.

In Fig. 2, it is found that cropping pattern C is the best for all the farmlands in terms of maximum total income generated. It is found that cropping pattern A generated the lowest maximum total income in all the 4 farmlands. In farmland 75 ha, the maximum total income for cropping patterns B and C are nearly the same. It can be concluded that cropping pattern C is recommended for farmers in the study area to generate maximum total income.

In Fig. 3, it is found that cropping pattern C uses the highest volume of total irrigation water of all the cropping patterns. In the farmland 70 ha, both cropping patterns B and C use the same volume of total irrigation water. It can be concluded that cropping pattern C can only be advisable if enough irrigation water is available otherwise, cropping pattern A and B will be preferred because of less irrigation water that will be used.

Fig. 8:	The total incomes in South African Rand (ZAR) generated from different strategies of DE for farmland 135 ha using cropping pattern C

In Fig. 4, it is found that in cropping pattern C, labour requirement is the highest in all the farmlands except farmland 135 ha where cropping pattern B has the highest labour requirement. In fact, in farmland 70 ha, the labour requirement for cropping pattern C is abnormally high. It can be concluded that cropping patterns B and C can only be preferred by farmers who have enough available man power on the farm. Where there are limited workers on the farm, cropping pattern A will be preferred by the farmers.

Since cropping pattern C is preferred to others in terms of total income generated, only cropping pattern C is studied to find the most suitable DE strategy of all the 10 strategies for the model in terms of total income generated. In Fig. 5, it can be seen that strategy 3 generated the lowest total income of ZAR 2 416 500 and strategies 1, 2, 4 and 5 generated total income of ZAR 2 475 300 for farmland 60 ha. It can be seen that all the strategies with binomial crossover method except strategy 3 generated the same total income. Therefore, binomial crossover method is the best for this model. Also, strategy 3 is the worst of all the strategies for the farmland 60 ha in this model.

In Fig. 6, it is found that strategies 1, 2, 4 and 5 have the highest total income of ZAR 2 729 500. Strategy 9 generated the lowest total income of ZAR 2 721 800. Also it is concluded from the results that strategy 3 is the worst strategy of all the strategies with binomial crossover and strategies with binomial crossover are the best for the model in terms of total income generated.

In Fig. 7, it can be seen that all the strategies with binomial crossover (strategies 1 to 5) generated equal total income of ZAR 3 200 100. They all perform equally in terms of total income generated. Strategy 10 is the worst with only ZAR 3 176 800 total income generated.

From the results in Fig. 8, total income of ZAR 5 325 900 is generated by strategies 1 to 5 on the farmland 135 ha. Strategy 7 generated the lowest total income of ZAR 5 221 700. In this Fig. , it can be concluded that strategies 1 to 5 with binomial crossover method are better than the ones with exponential crossover method (strategies 6 to 10) in this model.

In summary, DE is a good optimizer for determining crop pattern in a water scarce environment. Previous results on the application of DE to irrigation planning and crop planning show that DE strategies with binomial crossover are the best for the models (Reddy and Kumar, 2007; Janga and Nagesh, 2008). This is in support of this study that finds strategies with binomial crossover the best for determining the optimum cropping pattern. This study was able to determine the optimum cropping patterns that will increase production returns in the Lower Orange River catchments as proposed.

CONCLUSIONS

The 10 strategies of DE are applied to the cropping pattern model in the Lower Orange irrigation areas to determine the best strategy for the optimization model. The irrigation areas are divided into farmlands of 60, 70, 75 and 135 ha by the management of Lower Orange River catchments and distributed to farmers in these catchments. In this study, 3 different cropping patterns are studied to determine the best cropping pattern for the study area in terms of maximum total income generated from farming, irrigation water use and total man-days required on the farm. It is found from the results that DE strategy 2 (DE/best/1/bin) outperformed other strategies. Also, it can be concluded that DE strategies with binomial crossover method (strategies 1 to 5) outperformed DE strategies with exponential crossover method (strategies 6 to 10).

It can be concluded that if enough irrigation water is available for farming in the study area, cropping pattern C is the best for farmers to generate the highest total income. Enough farm workers must also be available to consider cropping pattern C. It is found from this study that cultivating larger area of land may not necessarily imply higher profit for the farmers. A small area with good cropping pattern can generate more profit. Therefore crop planning is essential in a farming business and this model is a good choice for cropping pattern in a water scarce environment like South Africa. The whole catchments can be further studied to find the optimum cropping patterns that will be suitable for each sub-area. Moreover, reservoir operation can be incorporated to the study to find appropriate release policy that will satisfy all the water users downstream of the dam including the farmers.

ACKNOWLEDGMENTS

Tshwane University of Technology is appreciated for providing necessary funds for this project.